Download - QUARTERLY OF APPLIED MATHEMATICS...In the first equation of (2.5), the square roots l\{x) and \/t\ [x) — x2 are to be taken as odd and even functions of x, respectively. The determination

$: QUARTERLY OF APPLIED MATHEMATICS...In the first equation of (2.5), the square roots l\{x) and \/t\ [x) — x2 are to be taken as odd and even functions of x, respectively. The determination$
QUARTERLY OF APPLIED MATHEMATICS

Volume LIX March • 2001 Number 1

MARCH 2001. PAGES 1-24

ELEMENTARY EVALUATIONOF CERTAIN INFINITE INTEGRALS

INVOLVING BESSEL FUNCTIONS

By

V. I. FABRIKANT (Cowansville Jail, Cowansville, Quebec J2K 3N7, Canada)

AND

G. DOME (formerly at CERN, CH-1211 Geneva 23, Switzerland)

Abstract. Although it is known theoretically that certain infinite integrals of Besselfunctions can be expressed in terms of elementary functions, the practical evaluationof such integrals was quite difficult due to the algebraic complexity of the expressionsinvolved. A simple and elegant algebra is introduced here which allows these integrals tobe calculated in an elementary way in terms of elementary functions. Some relationshipsare shown between the integrals involving Bessel functions and two-dimensional integralsover a circle of elementary functions involving distances between points. A comparisonis made with existing results, and some of them were found in error (or were misprints).

1. Introduction. The following infinite integrals are encountered in various engi-neering problems (see, e.g., Elliott [4]):

BK cossin

(■ax)Jv(px)dx. (1-1)

In the case of integer /z = m and integer v = n, the integrals (1.1) can be calculated inclosed form in terms of elementary functions. In practice, though, only a limited numberof such integrals is given in the tables. For example, in the table of Gradshteyn and

Received November 4, 1997.2000 Mathematics Subject Classification. Primary 33C10, 33C05, 44A10, 31B05.

(C/2001 Brown University

2 V. I. FABRIKANT and G. DOME

Ryzhik [5, No. 6.7513, 6.7521, 6.752 2], one can find

/Jo

(z2 + p2 — a2)'2 + 4 a2z2 + z2 + p2 — a2e~zx cos(ax)Jo(px) dx —^ (1-2)

\[2 (z2 + p2 — a2)2 + 4a2z2

[Rez > | Ima| + | Imp|],

00 dx 2 a \e zx sm(ax)Jo{px) — = arcsin I . ) (1.3)

x I y/z2 + (a + p)2 + sjz2 + (a - p)2 )I[Rez > | Ima| + | Irnp|],

[ e~zx sin (ax)Ji(px)— = - (a \- Jyj(z2 + p2 — a2)2 4 a2z2 — (z2 + p2 — a2)^Jo x p V n/2 v J

(1.4)[Re z > | Ima| + | Imp|],

where the latter formula is given in an implicit form.In principle, when v = 0, the integrals (1.1) can be evaluated from (1.2) and (1.3) just

by direct differentiation with respect to z for positive m and by integration for negative m.Practical implementation of these ideas, however, is quite difficult: differentiation wouldlead to unwieldy expressions, while integration seems to be too formidable to perform.Some elegant algebra is introduced below, which allows us to simplify these calculationssignificantly. A number of integrals of type (1.1) for integer parameters p = m and v = nare given in Fabrikant [6, p. 254].

2. Evaluation of the integrals. In order to overcome these difficulties, the followingnotation (Fabrikant [6, p. 12]) is introduced:

li (x) EE l\ (x, p, z) = i [y/(x + p)2 + z2 - ^/(x - p)2 + z2], (2.1)

h(x) = l2(x,p,z) = \ W{x + p)2 + z2 + yj(x - p)2 + z2] (2.2)

[x > 0, p > 0, z > 0].

If x, p, z are complex, the square roots in these formulae are to be taken with positivereal parts. The geometric interpretation of l\ and I2 is quite obvious if one considers acircle of radius x in the plane z = 0 and a point with the polar cylindrical coordinates(p,<f>,z); l\ and I2 then represent half of the difference and the sum, respectively, of thelongest and the shortest distance from the point to the circle.

The following properties of l\ and I2 can be verified directly:

lim l\(x) = min(a:,p), l\<m\n(x,p) for any z, (2.3)

lim hix) = max(a:,p), I2 > max(x, p) for any z,z—>0

INFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS 3

lUx) - ll(x) = V(x + P)2 + Z2s/(X- p)2 + Z2,

l\{x) + l2(x) = X2 + p2 + Z2, (2.4)

k{x)l2(x) = xp,

\!X2 — l\{x)yjl2{x) -X2 = xz, \JP2 - - P2 = pz, (2.5)

\Jx2 - lf(x)yjp2 - l\(x) = zli(x), \Jl2{x) - x2^Jl%{x) - p2 = zl2(x). (2.6)In the first equation of (2.5), the square roots l\{x) and \/t\ [x) — x2 are to betaken as odd and even functions of x, respectively. The determination of the other squareroots follows from the other equations (2.5) and (2.6).

The differentiation can be performed by using the main formulae:

d , (r) xl^x) - p1^x) _ p\x2 - li(x)] (97)dp 1 ll{x)-ll{x) h{x)[lj(x) - lf(x)\

x[l%(x)-p2] d= ~xZ 2(x)>l2{x)[l\(x) - /i(x)] dx

9 ! (t\ Pl2(x)~xh{x) p[lj(x)-x2]dp 2 l%(x)-l2(x) l2{x)[ll(x) - l\(x)\

x[p2-lf{x)\ d=l\ (x) [/^ (^) — 'i(^)] dx

d , . . zlAx) d , , \ zlo(x)d~zh{x) = ~ i2(X) - i2(xy d~z2^x) = wrwr

A number of additional formulae can be found in Fabrikant [7, p. 338].Using l\ and l2 in (1.2)—(1.4) allows us to simplify these formulae significantly. With

the abbreviations l\ = li(a,p,z) and l2 = l2(a,p,z) we have

fJo

fJo

fJo

//2 _ a2zx cos(ax)Jo(px) dx = —-pr~, (2-10)

2 ~ 1

zx sm(ax)Jo(px)— = arcsin ( — J = arcsin ( — ) , (2-11)x \l2J \pj

zx sm{ax)J1{px)— = a~Va2~li (212)x p

[Rez > | Ima| + | Imp|].

The simplicity of (2.10)-(2.12) makes it possible to differentiate or integrate whenevernecessary. For example, integration with respect to z of both sides of (2.12) yields

[«nwt - +1 arcsln C i) (2.13)x Zap 2 V 2/

[Re z > | Im a\ + | Im p|].

To illustrate the method, we show this in some detail. Indefinite integration by parts ofthe negative right-hand side in (2.12) with respect to z gives

/>OC

Jo

_ r d_a- ^a2-l\ = _a — \Ja2 — I2 + 1 f hdlJ ~ P P PJ ' y/a2 — I2


Using (2.6), we obtain for the integral on the right-hand side

1P .

J \Jp2 ~ li dli = Yp li \Jp2 ~li + P2 arcsin ^ —

hence,

J dz-—= ± 2a + 2^ja2-l\ + + £ arcsin ̂ .

Using (2.6) again, we finally obtain

, , a — \Ja2 — tr z 1- dz —jdZ- ^± = Yp^^ ~^a\Ja2 — l\ + 2(a2 — I2) + I2

2p v^Tf1 2 ( H, rt\2 P ■ (I ia — \laz — If I + — arcsm | —

, P ■ h+ - arcsin ( —

2^ yja2 -1\ \ V / 2 \ p

We note here that, by going from the indefinite to the definite integrals, the integrationconstant must vanish: as 2 —> oo, the integral in (2.13) vanishes, and also 1-2 —> z andl\ = ap/l2 ~ ap/z —» 0. Applying (2.4) and (2.5) finally yields (2.13). (Note that thearcsin-term in the corresponding formula [6, No. (5.3.16)] has the wrong sign.)

In the table of Prudnikov et al [11, No. 3.12.11.16], the integral (2.13) reads (aftercorrection, i.e., multiplication of the right-hand side by a2),

fJ 0\{ax)J\{px)^r — -^-[Ry/a2 + z2 sin(i9 + 77) — (a2 + z2) sin(2i?)]

x1 '2p

p ( R sin t1 + \/a2 + z2 sin \+ — arctan

2 \R cos 77 + \Ja2 + z2 cos $ J[Re z > | Ima| + | Imp|],

where

R4 = (z2 + p2 - a2)2 + 4a2z2, i.e., R2 = I2 - I2;

tan ■d = a/z, 2az cot(2ry) = z2 + p2 — a2.

Evidently, this formula is more complicated than (2.13). Note that the correspondingformula in Prudnikov et al [9, No. 2.12.25.6] is also incorrect: for p2 read p in thedenominator of the first term.

Quite a number of other integrals can be calculated by using the well-known recurrencerelation for the Bessel functions

Jn+i{z) = 2nJn(z)/z - Jn-i(z). (2.14)


For example, for n = 1 we obtain

f°° dr r°° ( 2 \ dr/ e~zx sin(ax)J2{px)— = / e zx sin(a:r) I—JApx) — Jo(px) ] — (2-15)

Jo x Jo \PX J x(2a2 — — a2 — 2 a2z

ap2

sjl\ - a2(a - \Ja2 - l\)2ap2

[Re z > | Imo| + | Im/o|],

[ e~zx sva{ax)J2{px) dx — [ e~zx sin(ax) ( — Ji(px) — Jo{px) ] dx (2.16)Jo Jo \PX J

2 (a — y/ a2 — l\) yja2 — l\p2 - l\

[Rez > | Ima| + | Imp|].

The integrals of higher-order Bessel functions can be obtained in a similar way. Inte-grals of this type can be found in Prudnikov et al [9]. For example, the integral (2.15)corresponds to 70 in [9, No. 2.12.25.3] for a = 0 and v = 2:

/Joe zx sm(ax)J2(px)— = P sin (2 arcsin f y

x 2ih + Vll~P ) V V 2[Re z > | Ima| + | Imp|],

which is another form of (2.15).These integrals of Bessel functions can be related to two-dimensional integrals over a

circle. For example, the following integral was given in Fabrikant [7, No. (A5.41)]:

/»27r p

Jo JOr27T r° Po dpo difo 0 ■ (a \ M= 2-n arcsin I — I , (2-17)0 Jo Ro\Ja2 - p20 V

where Rq = p2 + Pq — 2ppo cos(ip — <^q) + z2. Comparison of (2.11) with (2.17) allows usto write

o r -zx., ^ f2n r podpodw /oio,2tt / e sm{ax)J0{px)— = / / —— (2.18)Jo x Jo Jo Ro v a ~ Po

Many other relations can be obtained by differentiation and integration of (2.18).For a nonnegative integer v = n, the following formula was given by Fabrikant [6, p.

348] without a direct proof:

f°° dr r2~ rLl r2lJ dr/ e-zxJl/+1/2(ax)Mpx)^= = J-(ap)-» (2.19)Jo y/x \ 7T a JQ y?fP - x2

[v > - i, Re z > | Im o| + | Im p|] .

We prove it here by induction. For v = 0, this formula is correct by comparison with(2.11). Assuming that (2.19) is correct for a given v, we show that it is then also correct


for v + 1. Thus we shall prove that

r°° rlT I 9 Z"'1 t-2i/+2 rt-r/ e-*xJ„+3/2(ax)J„+1(px)^ = J — (ap)-^ (2.20)Jo Vx V na Jo yjp2 _ x2

Using the well-known properties of Bessel functions (e.g., Gradshteyn and Ryzhik [5, No.8.472]),

fJo

d_dp

J„(px) xJv+i(px)(2.21)

av+3/2 Ju+l/2(ax) da = au+3/2 Ju+3/2(ax)/x, (2.22)

and substituting y = x/p in (2.19) yields

fe-%1/2(am| =\[~{-Y fl/Pj7T^ (2'23)Jo Vx v 7Ta\aJ J0 J\-y2

©7V1 - v

"/h y2v dy

\Ji -y2from (2.4). Dividing both sides of (2.23) by pv, differentiating with respect to p, andtaking account of (2.4)-(2.8) leads to

sfxe zxJu+i/2{ax)Ju+i{px)dx=\j^[^j(l\ - a2)ap

- i\n

These expressions, based on a/l2, will be referred to later. For the present purpose, it ismore straightforward to use l\/p, which yields

fJ 0\fxe zxJu+i/2{ax)Ju+i{px)dx

'SOwa \aJ \p J - q

Y l i2"+1 p2 -1\

2"+1 - r p(a2-l2) + 1

m-m p\it a a»p»+1 yJfP- _ q

Applying (2.22) to both sides of this equation gives

a.+3/23/21 e~"J"+'<l{ax)J"+,ipx)^ = \fl7*L ^Tp1WJi!ada- (2-25)Using (2.8),

P2~llq-q

a da — l\ dl\ (2.26)

yields, after division of both sides by au+3'2, formula (2.20). The proof is thus completefor integer values v = n.


Formula (2.19) is, however, valid for any u. Indeed, when multiplied by cos(^<^), bothsides of (2.19) represent harmonic functions (this can be verified explicitly for the right-hand side; it is done in the Appendix), satisfying the same boundary conditions; theyare thus equal. These boundary conditions are [3, No. 8.11(9)]:

at z = 0 : V(p<a) = — + ^ — pv cos(i/<p) \v >-£1 ,y/2Y{v + 1K+1/2 2j

dV/dz = 0 \p> a];at z = oo or p = oo : V = 0;

at ip = 0 and ip = tt/v < 27r : dV/d<p = 0.

The condition < 27t implies v > whereas the integrals in (2.19) are convergentfor v > — In order to prove the remaining case — ̂ < v < we shall now show thatif (2.20) holds, then (2.19) is also true. First, we note that

J^+i(ax)] =xJfl{ax), (2.27)rP j

-pv / p uJv+i{px)dp=-Ju(px) (2.28)J OO ^

Taking p, = v + A in (2.27), we multiply both sides of (2.20) by a^+l, calculate thederivative with respect to a, then multiply by Using (2.8) and then (2.4)-(2.6),this leads to

u l2iy+2 f 2 _ /2c JI,+i/2(ax)Jr+i(px)y/xdx = a~v~3/2 p~~v~l 1 1

i/+l/2P'/+1 V^2~a2i2v+2 /2 72 'l2 2 'l

In order to apply (2.28), we now multiply both sides of this equation by p ", integratewith respect to p (a being a constant), then multiply by —pv. This yields

Because of (2.8),

12 - a2p _ f Pdp — hdl2; (2.29)

hence,

r°° fir , r1/^ i 1 /1— rr t / \ t / \ WX Iy jyj_1 /9 / ^ fjf° e~zx Jl/+i/2(ax)Jv(px)<^= = p»a»+"2 ±0 ^2" a/1 ~~ a2!% \'2

Ca/^2 {fry.-1/2 ^

JoBy noting that a/l2 = h/p, we see that the last formula is the same as (2.19), i.e., (2.20)where 1/ + 1 is replaced by j/. This completes the proof.

V. I. FABRIKANT and G. DOME

We note further that

rx „2i' ™2i/+l

I = + (2'30)In the case of integer u = n, the integral on the right-hand side of (2.23) can be

expressed in terms of elementary functions (see Prudnikov et al [8, No. 1.5.2.2]):

r°° dx/ e-zxJn+1/2(ax)Jn(px)^ (2.31)

Jo Vx

"(£)n. , arcsin —(l)n \h

. 2 na

2 (£)„7Ta (1),

vf ~ 12 na

(!)"

^ r(n+i)I>-fc) /ax "2fe

S r(n-t +j)r(n) U [n > 0]

arcsm/ /a\2^(lWa^2fc+1

h)-r-\h) Kijjiik [n > 0]fc—0

[Rez > | Ima| + | Imp|],

where we have used l\l2 = ap. Note that the last series over k consists of the first nterms of the hypergeometric function

x 2F, (l, 1; |;x2) =2' y

The integral (2.31) belongs to the class described in (1.1), since Jn+i/2 is a combinationof sine and cosine functions, and of negative powers of x.

Our aim is now to generalize formula (2.19). We can apply the formula given byFabrikant [6, No. 1.4.18], [7, No. 1.3.23] for integer values of v, i.e.,

, 2 !'•" '"i"> f'im"d f p'^dpo,,,. . m= (2'32)

with a variation cos (yip). The potential V at z = 0 is supposed to be known for p < a,with the boundary condition dV/dz = 0 for p > a, and vanishing at infinity. FollowingErdelyi et al [3, No. 8.11(9)], a potential at z — 0 and p < a may be taken as

roc

V(p,<p, 0) = / xu~v+2n J,j,(ax) Jv(px) dx x cos(vip)Jo

q 9 9 1 r (z^ -f- n 75) — 2iy—2n— I u v 2 '

r(i/ + i)r(/u - v - n + i)

x2F1 + n + ±, v - n + n + is + 1; ^ J x cos (vf)

[p > v + 2n — 1, v + n + \ > 0] .


Formula (2.32) allows us to generalize formula (2.19) or formula (2.23) to

/°° 1 / ft \ 11 — V — 2n— 1 / rt\Ve-zxX"-^+2nJ^aX)MpX) dx = — (-) (2.33)

1 ^ r(^ + n + q + |)(^ - n + n + i)q£T{n- v-n+ \) ^ q\Y{v + q+\)

rh/p tjT / 72 \ 1x a~21 / ( p2 + 2 ^

v I-*2[Re z > | Ima| + | Imp|,/x > v + 2n, n = 0,1,2,..., v > — i] .

The series terminates when p = ^ +2n+m (m = 0,1,2,...), and the integer n ensuresthat dV/dz = 0 for p > a in the left-hand side. The condition v > — | ensures that theintegral in the right-hand side converges at x = 0, and that the potential vanishes atinfinity. In this integral, the 5-power (including x2q) is the power of

t2=l\{t) (l+ . V x = h(t)/p, (2.34)p2 -l\{t)J '

where p and z are constant.Formula (2.33), which is obtained for integer v, can be proved to be valid for any

v by using the same argument as for proving formula (2.19). For v > g, a ratherlengthy calculation, given in the Appendix, reveals that each g-term in the series, aftermultiplication by cos(vtp), is a harmonic function. For — | < v < i, it can be shownthat from the assumption that (2.33) is correct for (p, + 1, v + 1) it follows that it is alsocorrect for (p, v). A derivation is given in the Appendix.

The series over q in (2.33) can be summed in closed form. We rewrite this relation as

fJoe-zxxv-»+2nj^ax)j^px) dx

1 1 aM-2*-2n-l p (i, + n+l)

~ ~p" T(p-u-n+ i)r(i/+ i)

fJody _y2„^{v + n+\)q{v-p + n+\)q

0 Vp2 - v2 +q=Q

where the sum now represents a Gaussian hypergeometric function. Using [10, Nos.7.3.1.4, 7.3.1.22], we obtain

*2 'r< I - . - . 1 . I ,12l2-Fi (^v + n+\,v- n + n+\\v+

' ( t2\2F1 ( )= 1-^

^2 \ /-i—v—211—1/2

(^-M + n+i)n / _ fN^"2"-172 „, / or, a. I. •> *2("+*) 2)n

1 ? 2^1 [ -n,p-n-,p-is-2n+±;1 -

= ( 1)n(fv (-")«(/*-")« A _ £_y-"-2n+q-1/2("+l)n ^ ^ (M - ^ - 2n+A), V c2


Thus,rOC

/ e-zxxu-»+2nJ^{ax)J„{px)dxJo

= 1 1 1 y\ (n-n)q a2n-2gy^7T 2M-^-2n-l aVpV Z_, yqJ ^ _ 2n _|_ q _)_ I)

[ll ,dy ,f,l2_i2r-u-2n+q-l/2Jo V>2 -r

[Re 2 > | Ima| + | Imp|, p, > v + 2n - |, n = 0,1, 2,..., v > — |] . (2.35)

Taking the derivative with respect to z, we obtain

fJo- zx v— J^(ax)Jly(px) dx

1 1 1 V( nn-qfn>\0f2/1-,-2n-i ^ \qjY(n-v-2n + q-±)

a2n-2q

x 221,2 \v2^+v - t2r-»-2n+i-v2Jo (p- - y r

[Rez > | Ima| + | Irn p\, p > v + 2n + \,n = 0,1,2,..., v > —1], (2.36)

For n = 0 and p = u + formula (2.35) immediately reproduces (2.19). For theevaluation of the integrals in (2.35) and (2.36), with (2.34) the following factorisationmight be useful:

p — y J p — y

3. Discussion. It is of interest to compare (2.19) with existing formulae in the tables.For example, Erdelyi et al [3, No. 8.11(21)] give

f°° dx/ e-zxJ„+l/2(ax)Mpx)

Jo \fx_ pvav+xl2Y{y+\) [* (sin<p)2»+ld<p

JJ 0\Z2-kT{v + 1) Jo [{z + ia cos ip)2 + p2]v+1/2

[Rej/ > -i,Rez > |Ima| + |Imp|] . (3.1)

It is not difficult to show that the case v = 0 agrees with (2.19):

a — iz\ . (a + izarcsin + arcsm

[Rez > |Ima| + | Impl], (3.2)and that the right-hand side agrees with (2.11) or (2.19), because

. (a — iz\ . /a + iz\ n (a \ .arcsm 1 I + arcsm I I = 2 arcsm I — I . (3.3)... ... M

For v ^ 0, the integration becomes quite complicated.

fJo

INFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS

Another formula is (Erdelyi et al [2, No. 4.14(24)])

(jnr

e~2X J„+i/2 (ax) Jv (px)\JX

c\ — 2v-L j. \£ti/ ~r i»_li /o 11r ./ i!/2r(2^ + i) „+1/2 .. ..

3 —a ' p [z + i(a + p)\r(i/+|)r(i/ + i)2

x i^2 ( 2z/ + 1, j/ + 1, i/ + i; 2i/ + 2, 2v + 1;1 ^ ~ „ 2 ia 2ip2u + 2,2z/ + 1; ——————-, ———————r

z + i{a + p) z + i{a + p)

[Re v > — Re z > | Ima| + | Imp|] , (3.4)

where (Gradshteyn and Ryzhik [5, No. 9.180 2])

= IN + W < 1]j=0 k—0 VCMCM J-K.

is a hypergeometric function of two variables. Prom (2.31) and (3.4) it follows that i*2can be expressed in this case in terms of elementary functions.

Prudnikov et al [11, No. 3.12.15.31] give the following integral:

fJor -z, T, w , w /2 a"p^1'2^2"-1 y/qs/xe Jv(ax)Jv+1/2(px)dx = ^ rr—

V 7T x/(p + a)2 + z2J(p -y/(p + a)2 + z2sj(p- a)2 + z2

[Re v > — 1, Re z > | Ima| + | Imp|],

which is equivalent to (2.24) when v is replaced by v+ or to the z-derivative of (2.19).A very general integral can be found in Prudnikov et al [9, No. 2.12.38.2] or in Erdelyi

et al [3, No. 8.11(22)], where in the latter the first factor given in (3.5) is missing. In [9],two expressions for this integral are given in terms of hypergeometric functions of twovariables F2 and F4 [5, No. 9.180 4] respectively, and a third one as an infinite series ofGaussian hypergeometric functions 2^1- Expressed in terms of 2-Fi, the integral reads

fJo

hvc>xa~1e~px Jfi(bx) J„(cx) dx

2M+"pa+M+^r(^ + 1)

r(a + p + v + 2k) f c2 \ I u^ k\T(p + k + l) 21 (-k'-V-Kv + lip)

[Re(a + p + v) > 0, Rep > | Im b\ + j Im c|]. (3.5)

This formula can also be deduced from Watson [12, Sect. 5.41(2)]. Watson [12, p. 399]gives also another form of (3.5), which in our notation reads

fJoxa e px Jfi(bx)Ju(cx) dx

(^b)^(^c)1' ^ r(a + p. + u + 2to)r(i/ + l)(c2+p2)(Q+^)/2 ■i-^0 to!P(/x + to+1)

_ ,'a + p + v a + p-u- 1 c2 \ ( b2 .+ m' 2 m'^ + h^T?)[~WT7)] ■ (3"6)


We observe that the hypergeometric function 2-fi in this equation may be written as

, + a + ft + „+1 #) *F<{ ^ +m> 2 +

Thus, by writing 2F1 in (3.5) explicitly as a series over n and letting k — n = rn > 0, itis easy to see that (3.5) and (3.6) are equivalent.

Prudnikov et al [9] give quite a number of useful special cases for the integral (3.5),whereas the general formula seems to be of little use in numerical applications. Inparticular, when p is small, difficulties of convergence are likely to occur in (3.5) if p < b.

By comparing (2.24) with (3.5), we find an interesting summation formula:

av~l/2pv 0—2^+1/2--2^-1 r(2i/ + 2k + 1)

I>+1) ^k\T{v + k+\)

I j p2\ ( a2 \ k [~2~ (h\v \Jl\ —x 2^1 ( ~k, \ - v - k;u+ 1; — ) ( -- ' ' '4 z2 J V it a \h J I2 — l2

The following notation is used in Prudnikov et al [11, No. 3.12.11]:

r = \[V(a + p)2 + z2 + v/(a- p)2 + z2]2, (3.7)

R4 = (z2 + p2 — a2)2 + 4a2 z2, (3.8)

tani? = a/z, (3-9)

2azcot2rj = z2 + p2 — a2, (3.10)

w± = 11\/{z± ia)2 + p2, (3-11)

v± = p~l[z ±ia-\- \/(z ± ia)2 + p2], (3-12)

w± = —L-\/~\/(z2 + p2 - a2)2 + 4a2z2 ± (z2 + p2 - a2), (3.13)\/2a v

(p = arg(z2 + p2 — a2 — 2iaz) (—7r < <p < n), (3-14)

ip = v arcsin(2a/r). (3.15)

Note that 2r/ = —(p; hence, one of the definitions (3.10) or (3.14) is superfluous in anycase. Note also that (3.15) for ip is incorrect. The notation r in (3.7) corresponds to I2-,so the correct formula for ip should read

i/j = i/arcsin(a/Vr), (3.16)

which corresponds to ua,rcsm(a/12)■ Thus, introducing just two parameters l\ and I2simplifies the notation (3.7)-(3.16) considerably and effectively eliminates the need forthem in all cases. For example, using l\ and I2 in (3.8) leads to a complete square, i.e.,R4 = (l22-lj)2.

Prudnikov et al [11] also use

^{p + v+lp + v + 2 | i p2 ^z±-(z±ta) 2*i 2 , 2 ;i/+l; ^ ± ,^2 J . (3.17)


Defining x, y by the two equations

■ , / , z + ia z — iasmh(x + iy) = , smh(x — ly) = (3.18)

P Pfor complex z,a,p implies

sinhxcosy = z/p, cosh x sin y = a/p. (3.19)

The simplest solution is given by

coshx = l2/p = a/li, s'my = a/h = li/p [Rex > 0]. (3.20)

Let now u = x ± iy be such that

z it idsinhu = [Reu > 0], (3-21)

where one sign is chosen. This equation has two solutions: u and (—u±in) in eu or e~u.Imposing the condition Reu > 0 makes eu and e~u uniquely defined. To enforce thecondition Reu > 0, take Rex > 0 and Re(±iy) > 0 in (3.20). This makes ex,e±ly, andeu uniquely defined. If Re(±iy) = 0, i.e., if y is real, (3.20) cannot distinguish betweeny and 7t — y; the choice is made by the sign of cosy in (3.19) when Rex > 0.

Using (3.21) and a quadratic transformation of the hypergeometric function [10, No.7.3.1.56]:

fp + u+ lp + u + 2 p22F1 —~^—;^ +1;-

2 ' 2 ' {z±ia)2/= (l_e-2«)^+12Jp1(^ + !/+i,/i+i;I/+i;_e-2«) [Re tx > 0], (3.22)

the relation (3.17) can be written as

(2e~u \ 'l+iy+1z± = f —— J 2Fi{p, + v + l,p,+ l;is+l;-e~2u).

When z > 0, a > 0, p > 0 are real, then

u = x ± iy, (3.23)

coshx = I2Ip = a/l\ > 1 [x > 0];siny = a/l2 = h/p < 1 [cosy > 0, i.e., 0 < y < 7r/2].

Using 11 and l2 with (2.4) and (2.5) in (3.11)—(3.12) allows us to separate the real andimaginary parts in a simple manner:

u± = [y/(z± ia)2 + p2)^1 = \Jl2 — a2 ± i\ja2 — 12 (3-24)

= [Ree^^^2]-1 = (pcoshu)-1,

v± = — a2 — l^j I2 — a2 ± ia

ap

(a — \Ja2 — I2) y/l^ — a2 ia)RE = ex±iy _ e


The expression for w± in (3.13) can also be simplified:

sj% - a'2 \Ja2 -l\w+ = , W- = —

a aThe argument of if in (3.14) is transformed into a complete square:

2

f = arg (V'i-a2 "iyja2 - i?) ,

-isja2 -Z?) .| = arg ( \//| - a2 - i\/a2 - l\ ) . (3.25)

In (3.25), the two square roots are even and odd functions of a, respectively.Apelblat [1, p. 246] had introduced quantities A and B which are the same as w+

and defined later as (3.13) by Prudnikov et al. The notation R,tp is also used inGradshteyn and Ryzhik [5, No. 6.753]. Several examples are considered below in orderto show how the new notation allows us to simplify results given in the tables.

The following integral is given in Prudnikov et al [11, No. 3.12.11.12] (without thefactor a3 in the denominator):

/J 0

. T, . , 3w+w^(zw + + aw-)-aw% - zw3xe smaxJoipx) ax = 5-—~ ^5

' -,3(^,2 1 nn2. ^3a3(w+ + w_)3

[Re z > | Ima| + | Imp|]. (3.26)

The complexity of (3.13) obscures the obvious fact that w+w_ = z/a. The same integralin our notation reads

fJo-zx . r, °2Vl2 - a2[2a2{a2 + z2) - l\(a2 + p2 + z2)}

xe sin axJo(px) ax = a{ll - If)3[Re z > | Im a\ + | Im /?|],

Yet another example is taken from Apelblat [1, No. 12.4.72], which in a modified notationreads

fJ 0

1 -zxn \Tt \dx a2(l - B) + zaA - ZyJ z2 + p2e zx(l - cosax)Ji(px)^ = — xz 2p

+ yj Z2 + p2(a+£) + (b + 1)= - Ina

[Rez > |Ima| + |Imp|]. (3.27)

The first logarithm in (3.27) is a complete square which can be combined with the secondlogarithm, and the whole expression can be simplified significantly to read

fJ 0

dxe~zx(l — cos ax)Ji(px) —

12p (a - \Ja2 — + z (^Jll - a2 - \/p2 + z2^j I P\nl2 + ^l2~p2

2 Z + yj Z2 + p2

[Rez > | Ima| + | Imp|].


The integrals in (1.1) can be represented as the real and imaginary parts of the integralr O©

I£= e-^z-ia)xJu{px)x'1dx. (3.28)J 0

For /i = 0, Gradshteyn and Ryzhik [5, No. 6.611 1] give

' dx = (3.29)p" y/{z- ia)2 + p2

exp{—v arc sinh()}

\J(z- ia)2 + p2[Re v > — 1, Re(z — ia) > | Im p|].

The arguments of the numerator e~vu and of the denominator in (3.29) are given by(3.21) or (3.23) and tp/2 in (3.25), respectively.

Using the notation li and I2 in (3.29) allows us to separate the real and imaginaryparts in a simple way:

fJ 0

fJo a"pu(\/l 2 — a2 — i\/a2 — I2)

[Re v > -1, z > 0, a > 0, p > 0], (3.30)

Prom (3.23) and (3.25), the arguments of the numerator and of the denominator are vyand <p/2, respectively.

The case of a positive integer p = m in (3.28) can be calculated by differentiation of(3.30) with respect to 2. The case of a negative integer p = —to requires integrationwith respect to 2. Both operations can be performed in a simple way by noting that

ddz

a ± Ja,2 - I2 ) [Jl% - a2 T (a ± y/a2 - lj)(\/l2 ~ a2 T ia)

\Jl\ —o2 — iyja2 — IfIntegration of both sides of (3.30) with respect to z yields

fJo' e-(z-ia)xj (px)te = (« - Va2 - llYWl'i - Q2 + ia)1

" x vaup"

[Re v > 0, z > 0, a > 0, p > 0]. (3.31)

Formula (3.31) agrees with Gradshteyn and Ryzhik [5, No. 6.623 3]. A similar result inPrudnikov et al [11, No. 3.12.11.10] has a misprint: the parameter 1? (as defined by (3.9))should read ip (see (3.16)).

Formula (3.31) can be integrated again with respect to 2:

fJ 0)xMpx)^ = fxz 2 v

fu-l ^ f„+lZ/-1 V + l_

[Re^ > l,z > 0, a > 0, p > 0], (3.32)

where

y = (a ~ Va'2 ~ l2i)(\/l2 - a2 + ia) = e-a /3 33xap

and where sinhi* = (z — ia)/p, Reu > 0.


Formula (3.32) agrees with Prudnikov et al [11, No. 3.12.11.11]. Prom the structureof (3.32) it is clear that (3.31) can be integrated with respect to 2 as many times asnecessary. For example, a third integration gives

I °° At n2e-(z-ia)Xj^ = E-X6 \v

1 / jv—2 fU\ 1 (fv fvJr^

v — 1 \ v — 2 v J v +\ \ v v + 2 /[Ref > 2,z > 0,a > 0,p > 0]. (3.34)

There seems to be no equivalent of this formula in the tables. The imaginary part of(3.34) remains finite when v = 2, but the result should be calculated as a limit for v —> 2.

There exists a general result in Gradshteyn and Ryzhik [5, No. 6.6211] and in Prud-nikov et al [11, No. 3.12.11.5]:

rOC

/ e-{z-ia)xJ„(px)x»dx

pl'r(At + f + 1) //u + ^+ l/x + z^ + 2 p2"{z - iay+v+lT{v + 1) 2 2 2

[Re(p + v + 1) >0, Re(z — ia) > | Imp|]. (3.35)

Using (3.22) with sinhit = (z — ia)/p, Re it > 0, we obtainrOO

/ e-<z-ia^xJiy(px)x"dx (3.36)Jo= " e~2ur+V+12F\(p + v + l,p+l-,v + l-, -e"2")

= (2) r+"+12Fi(p + ^+ 1^ + 1-^ + 1;-f2)

= (2)T{r(tlV)F+,y+1{1 + f2)~2"~^Fl(_/x'" - " + *;-/2)[Re(/i + v + 1) >0, Re(z — ia) > | Im p|],

where e~2" = f2. The hypergeometric functions in the last two lines of (3.36) arepolynomials when p = —n — 1, n = 0,1,2,... and p = n or p = v + n, respectively.

One may verify from the comparison of the previous results with (3.36) that thehypergeometric functions in (3.36) can be expressed in terms of elementary functions forany positive or negative integer p. Thus (3.31), (3.32), and (3.34) are special cases of(3.36) for p = — 1, p = —2, and p = —3, respectively. It will also turn into elementaryfunctions for any p = v + n, where n > 0 is an integer.

For example, for p = v + 1, the last line of (3.36) gives

fOC/ e~^z~ia,s>x Jy{px)xv+l dx

Jo= -^=22l/+1r (1/ +§) f2u+2( 1 + f2r2"-\ 1 - /2)

2

= —=2u+lY (v + pu{pcos\\ u) 2v 3(psinhu),V71"


because of f = e~u. With (3.21) and (3.24), it may be written as

r ^ +Jo ~ °2 ~ i\Ja'2 l\)2"+3

[Rei/ > — l,Re(z — ia) > | Imp|]. (3.37)

In this particular case, the same result can also be obtained directly from (3.35) whenusing (3.24). It is in agreement with Gradshteyn and Ryzhik [5, Nos. 6.753 3, 6.753 4],apart from a misprint (read instead of /„), and with Prudnikov et al [11, No. 3.12.11.8].

The separation into real and imaginary parts in (3.37) can be achieved by rewritingthis formula as

fJ oe-^-ia)x Jv{pX)Xv+x dx

2(2p)T(i/ + |)(z - ia)Mi22-iir+3/2 exp

2(2 pyY{u+\)(z-ia)

a2 —12i(2v + 3) arctan J ~2

\ to CL

~ l2)u+3/2exp i( 2v + 3) arcsin < la2 ~ f?

I2-I2

[Re^ > —1,2 > 0, a > 0,/O > 0],

where

a2 — li ifarcsm y = " 2

is defined in (3.25). Numerous other examples can be considered. This is left to theinterested reader.

4. Conclusions. The introduction of the parameters l\ and 12 according to (2.1)and (2.2) allows us to simplify significantly the calculation of various infinite integralsinvolving Bessel functions. It eliminates the need for numerous parameters (Eqs. (3.7)to (3.14)), which are used in various tables of integrals. It seems appropriate to rewritethe integrals in the tables in a unified form, utilizing the parameters l\ and 12. In thecase of complex integrals, the introduction of these parameters allows a simple separationof the real and imaginary parts. Some of the integrals given in the tables are found tobe incorrect, and the corrected version is presented. The simplicity of integration anddifferentiation of expressions involving the parameters l\ and I2 allows us to calculatevarious integrals not given in the tables, for example, the integrals (2.31) and (3.34). Ananalogy has been pointed out between the integrals of type (1.1) and two-dimensionalintegrals over a circle, with the integrand involving distances between two points.

5. Acknowledgment. We would like to thank Dr. K. S. Kolbig (formerly at CERN)for useful remarks and comments, and for improving the presentation of the paper sig-nificantly.


Appendix. In this Appendix, we shall prove that (2.33) is valid for any v > — i. Westart with the case v > Here we shall prove that each g-term of (2.33), when multipliedby cos(v<p), is a harmonic function. In cylindrical coordinates, a function V(p,tp,z) isharmonic if

AT, 1 d f dV\ 1 d2V 82V n~pdp V '~dp ) + ~d^2 + -g~2 ~ °-

In particular, for a function V(p, tp, z) = pv f(p, z) cos(is, ip), we obtain

AV = <i -P

= P

y-'f + W + (* + + '"+'0 2 v-2f , vd2f\ ( n~VP f + P cos(M

2^ +Id/ d2f d2f cos(i/<p).p dp dp2

The condition AV = 0 is thus equivalent to

We shall show that each q-term in the series (2.33) satisfies (A.l). Therefore, we take

where the upper limit of integration x = a/l-i is a function of (p, z). We obtain

dfg _2p X2^2" ( 2 , Z2t; o p +

^2 a-2q ( a\2lJ + 2q ( n2 + l2 ( a^2u

dx sjl - x2 V 1 - x1 / \ 2^+2(3'

_ ^2 -r2^^ . _ yjl\ - a2 \h J V li-a2J sjl\ - a2

where from (2.6), p2 + z2l\!(l\ - a2) = IThus

1 dfq I -2uq _ 2i/+i

P dp y/q - a2l^-O2! -■>» rX dx /o z2 \q 1I212-I2

- 2^+1,-2^-2^2 "Z + -2q f1 <ir / 2 , '' Vll-ll J» vT^ 2"{P 1 — )

= —a


d2fq _ ddp2 dp

(id_U\P dp

= + p{-a2"+%2»~2 ^ ~ 0,2pdp n *2 q-q2v + 2 P(l2 a") , P 2p 2 ,72 9 2\

;2 ;2 ;2 "1" ;2 ;2 /;2 ;2\2^ 2 ' ll >l2 l2 ll l2 '1 ll2 'U

hence,

2v + 1 dfq + <92/?p dp dp2

"(2"+2» { -«2"+li""2 W + 2?a" I 7fe^+2' (^'+ ii)''

+ p2 \—a2u+ll j~2u—2 V^2 °2Z2 - I2

2u + 2l2 ^ + 1 2 n2 + j2 _ 2 2\ ,;2 I2 _ ;2 ' /2 72 /i2 ;2\2 "2 ^ H ^ >2l2 2 'l l2 'l vl2 n J f2j

/■x r/r / ~2 \9~2>

+ 4,(, - 1) Jo ^=^^+2Q ( P2 +x"2 V" 1 — a;2

a/, r2"2 „2i/+l

^ V 'l"*?a-2, r dx 2u+2Q2 fp2 _jM

Jo yV l"*2/2 \9"1 z

1 -X2

= "°2 V#tP + 2««"2' f ^t=i2"+2« ( / +^2 _ 'l io Vl - X2 V 1 - X2 / 1 - X2 '

52/-3 _ 2vi — 2u Va2~lidz2 ~ 2 Z2-Z2

2i/ T7y 177 +Z 1 zq 2z(/2 + /?)

q _q 1 a2_q q _ q (;2 _ Z2)2 J

Ilo2u a2"+1 / z \ „ _2o fx dx 2l/+2n+ qzjq^W^-\ W^J+ qa I 7^xZ2 \g"2 Z2 ( o z2 \q~1 1

2(9-D(p2 + t^o 7^-3^+ P2 +1 — x2 J (1 —x2)2 \ 1 — x2 J 1—x


By noting that from (2.5), \Ja2 — I'f = az we obtain

A/,) - '2'+V-2^f^-(2„ + 2) + (2» + 2&%=4 - J* + **<11 + I? - 2a2) - 2,£

, 2^+1/ — 2v — 2 ^ ^2

2 sjq^q-q

q q~q q-q (q-qyK2 1 ~ ;

2 r 2^ P2 — q 1 2(i|+if) 2q,i§-z? z2 zl-z? {q~q)2 q~a2_

2 \ <?_1

vn)Th^+2i?i+T^r»2

I dx o„^o„ / o Z2 V ' 1+ 2qa~2q / X2l/+2|? p2v/1 - a;2 V I - x2 J 1 - x2

2 \ 9-2

Trh1'""" + {l-x2)2'

We gather now all terms independent of q (they correspond to <7 = 0):

4 = a2^-2^-2 1 sjq^tq -qy

x |-(2i/ + 2)(ll - a2)(l2 - l\) + {2u + 2)^(^ - a2)2 - p2{l22 - a2)

+ 2p r(i| - a2)(q + l'i ~ 2«2) + 2vz2ll - l\{p2 - If) +2 r2/2 11iF^+<■>]}■12 -12_ 2 li

We note that Z|(p2 — '1) = '2 P2 ~ °-2P2 and combine the terms in the two square bracketsin ^4:

^(f* - «2)(l2 + i? - 2«2) + 2"3 " + 1?)fc2 ^2 fcl

= 2_ji ^[p2(^2 + '1 ~ 2a'2) + (ll ~ p2)(l2 +'?)]^ [p2(li + lf- 2a2) + (Z2 - p2)(Z2 + if)2 '1

^^[i2(i22 + i?) - 2 qq\ = 2/2(i2

We obtain

; V^l 24 _ _2i/+1i-2i/-2^

2 (if-i?)2

X i(2v + 2) _(i| ~~ if) + J2V2 ~ fl2)2

- p2 + 2v{l\ - p2) - p2 + 2/2

By noting that — (if — Z2) + (i| — a2)p2/q = — q + p2, the terms in the curly bracketreduce to {—2(Z2 — p2) — 2p2 + 2q} = 0. Thus the term for q = 0 yields a harmonicfunction, what was stated for the integral (2.19) or (2.23). We are left with

£(/,) = —2qa2u+ll


Vl2 - a2 I" P2 , ll - p2i — 2^ — 2'2/| - q ll % -

=x2"+2<> ( p2 + ^.2 \ 9-1

71^2 v i -.2 \ 9-2

+ 4g(^ + l)a 2q fJo

+ 4,(,-l)«-V/ (1_^)3/3*'"""(> + i-f^) (l-«2)

+4, i, - «-2' rd^^+2< (p'+

■x / 22 \ 9-2

2; 70 (1-x2)3/2 v 1-x2

-4,(,-l)«-V| (1 _^)3/2

Using the binomial theorem, we get

r2"+2<? I p2 +1 — x2

r(f)~ 2nn2v+lr2v~2^^2 °2 ^2 ^1^2M/9j- i2 Z2(/2_a2)

+ 4^+1) V f9 'V, 2V2(9-1-m)22m r ^L^„2„+2,771 / ^ (l_x2)m+l/2

- 4q(g -1)5] ~ 2) a-2V2(<?"1_m)^2m [mi=(i \ m / Jo

X'

"X x2i/+2g+2

m=0

9-1

A, (1 — X2)m+3/2

+ 4o ^0 — ~~ 1^a"2<?£i2(<?"1~Tn)22m f — x2"+29V mja " Jo (1 ^2)m+3/2

The last two integrals can be transformed into the first one by using

2/Jox2v+2q+2

Jo (l-X2)m+3/2-

_ _J_/1 _ r2\-m-l/2 2i/+2g+l _ 2(^ + g + j) /*'' 2v+2qm+ ' m+i Jo (l-x^+i/2x

and2 r dx x2»+2, = 2 fX dx f x2 \ +2,

Jo (l-x2)™+3/2X *JQ (1-^+1/2 ^l_x2

(1 - x2)-™-l/2a.2V+2,+ l _ [-i + l + i _ J 2 rm + 5 J Jo

1TO+ |

dx x2u+2q'0 (1-X2)m + V2

Noting that

we see that the terms containing the last remaining integral combine into

4r/9 " 1V-2V2(9"1_m)^2m f 7 \ m J Jo (l-x2)m+1/2

(1/ + 1) + (g - 1 - to) —T ( 9 ~ 9 ) 771 1TO+75 V 2/ \ m+i

= 0.


We are finally left with

£(/,) = -2 ga2"+V-2 + B,Vl2 ~ a

where

B = 2qJ21 1 /_ i\ r / 1"(9 - 1 - m) + I <7 - -

m=0

9-1m

X ^-(1 - x^-rn~l/2x2^+2q+la-2qp2(q-l-m)z2mm + 2

= 2gx2u+2q+1 a~2q ——^—- £ (9 19-1 /„ 1\ / .2

2(g-l-m)

\/l - Z2 V, m J \l-:rm=0 x 7 x

2 \ 9-1= 2<?x2"+29+1 a"29 1 f p2 + -1—,) (where z = a/l2)

Vl — x2 \ 1 - xz J

= 2qa2lJ+l —j==—]-2u-2i— l2

and hence C(fq) = 0.We now treat the case — | < v < We assume that (2.33) is valid for (fi + 1, u + 1),

i.e.,/*°° 1 / a \ fi — v— 2n— 1 /n\^+l

J e~zxx"~'* nJ^+i(ax)Jl/+i(px)dx= -j= ( —J (£) (A.2)

1 y-v I> + n + \ + g + l)(i/ + n + ^ - /x)gr(/i-^-n+|)^ g!r(i/+ ± + g + l)

a/'2 dx ( o z2 Vx a_2<?

, _dx_ 2„+2q+2 ( 2 z2 yJo VT^ [P+1-X?)

We apply (2.27) and obtain/•oo i

_/ e~zxxu~^+ +l JIM{ax)Jv+\{px) dx = — (A.3)

1 ^r(z/ + n+ i+ g + l)(i/ + n + £ - /z),X r(/z-^-n+i) ^ g!I>+| + <? + l)

x (^+12 (p,-u-n—h — q) a2M-2,-2n-2<?-2 dx_x2u+2q+2 / 2! Jo vi-x2 V

*2 \ 9

1 — X

v-\-l „2fj,—2v — 2n—2q—l j2qCL to ■ ,

a/1 - {a/h)2 V;2

\ 2^+2(7+3a \ 1a

\ a2-ij]

The curly bracket in (A.3), which contains all the relevant variables, can be written as

j-2p»+1 (v + n+±-p + q) a2M-2,-2n-2,-2 j* _^_x2u+2q+2 ^2 +

, nt/+ln2M-2w+l 1 1 ^2 q2 Ip i/q^21?+2 % - if \ -


We now apply (2.28) to this bracket and obtain

p" r dpp~»{---}J oo

2p dp (y + n + \ - p + q) a2M-2^-2n-29-2

rt dx ( ■ z' yL -ji^ r i-*v-LPa^n+ 1_J

U /To o i2i/ + 2 /2 >2 r iy/q^2l22V+2 l\-l\

= p»Uv + n+\-^q) a2M-2.-2n-29-2_J_

\9+1 d(a/l2) / a ^ 2i/+29+2

y/1 - (a//2)2 V2

m ^ 2l/+2g+2/ 2 z2 r* d(a/i2) (ai\ ,2q+2Jo y/l - x2 \ 1 - x2 / y0 ,/l _ (a/12)2 V2 /

rJo_j_ a2^i 21/ 2n / * T2|ydx

\ZT— X"where we have used (2.29). In the first integral, the variable is a (with p and z constant),whereas in the last two integrals the variable is p (with a and 2 constant). We notethat, when p —> 00,12 —> 00 such that Z2/P —► 1; therefore, the first integral is a definiteintegral that vanishes when p —> 00. In the last two integrals, the limits of integrationcorrespond to p and p = 00. Thus

r dpp-v{.-.}j 00

- p"g^-^-2n (i/ + n+ i~M + <?a-2q-2 f'2 dx ^2t/+2q+2 /^2 + ^ N ?+

9+1 Jo \/l - Z2 V 1 - ^,

y + n+i —u—1 x2udxI-M-1 P+1 7o9+1 Jo a/1 — :

Applying (2.28) to (A.3) gives

J™e-~x»-^Max)Mfx)dx,-L (I)"-"-2-' (£)"

if r(i/ + n + | + q + l)(t/ + n + | — /x)g+ir(^-i/-n+i) (<? + 1)!I> + | + g+ 1)

x a~2^ r , dX a^+^+D (V + ^i /'2io vT^~ V ' i-*:r(i/ + n + \ + q + l)(v + n - \ - p)q+i fh x2u dx

(<? + l)!T(i/+ \ +q+ 1) Jo \/l — x2


where the last integral does not depend on q. We see that the last series over q can berewritten as

+ n + \ + q)(u + n - \ - fi)qEq= i q\T{v+±+q)

T(i> + n +!>+ 5)~

I> + n + i)T(n+i)

r(i/ + n + \)Y{v+\)

Thus we obtain

[2F\ (is + n + v + n - \ - fi; v + l) - l]

- 1r> + -1/ - 2n +1)F(-n)r(/i - n + 1)

e-*x1 / n \ n — v — 2n—\ / n \ i

xu^+ "Jll{ax)Ju(px)dx = —= (-J (-J1

r(M — v - n+ |)

I> + n + \ +g)(v + n+ i -/z)q ^2(? n2 da; 2l/+2q / 2 z2g!r(i/ + i + g) ° /, vT^X V

which is the same as (A.2) where (/x + 1, ^ + 1) is replaced by (/x, i/). This completes theproof.

[1[2

[3

[4

[5

[6

[7

[8

[9

[10

[11

[12

ReferencesA. Apelblat, Table of Definite and Infinite Integrals, Elsevier, Amsterdam, 1983A. Erdelyi, ed., W. Magnus, F. Oberhettinger, and F. C. Tricomi, Table of Integral Transforms,vol. I, McGraw-Hill, New York, 1953A. Erdelyi, ed., W. Magnus, F. Oberhettinger, and F. C. Tricomi, Table of Integral Transforms,vol. II, McGraw-Hill, New York, 1954H. A. Elliott, Axial symmetric stress distributions in aeolotropic hexagonal crystals. The problemof the plane and related problems, Proc. Cambridge Phil. Soc. 45, 621-630 (1949)I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th edition, A. Jeffrey,ed., Academic Press, New York, 1994V, I. Fabrikant, Application of Potential Theory in Mechanics. Selection of New Results, Kluwer,Dordrecht, 1989V. I. Fabrikant, Mixed Boundary Value Problems of Potential Theory and their Application inEngineering, Kluwer, Dordrecht, 1991A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. 1, ElementaryFunctions, Gordon and Breach, New York, 1986A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. 2, Special Func-tions, Gordon and Breach, New York, 1986A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. 3, More SpecialFunctions, Gordon and Breach, New York, 1990A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. 4, Direct LaplaceTransforms, Gordon and Breach, New York, 1992G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., University Press, Cambridge,1966